Ok so this the problem given in my text boo
The principal value of $\sin^{-1}(\sin\frac{4\pi}{3}) + \cos^{-1}(\cos\frac{4\pi}{3})$ is
(A)$\frac{8\pi}{3}$ (B)$\frac{4\pi}{3}$ (C)$\frac{2\pi}{3}$ (D)$\frac{\pi}{3}$
and the following condition are given with it for $$(x<0 )---\{\frac{-5\pi}{2}\le\sin^{-1}x<0,$$ $$\frac{\pi}{2}\le\cos^{-1}x\le\pi$$
and for
$$(x\ge0 )---\{0\le\sin^{-1}x<\frac{\pi}{2},$$ $$0\le\cos^{-1}x\le\frac{\pi}{2}$$
However the condition were making no sense to me so i tried to solve the problem simply like this $$\implies \sin^{-1}(\sin\frac{4\pi}{3}) + \cos^{-1}(\cos\frac{4\pi}{3})$$ $$\implies \frac{4\pi}{3} + \frac{4\pi}{3}$$ $$\implies \frac{8\pi}{3}$$ (and those little disgrace gave this answer in the option too), but this option is wrong, the answer is D.
So were I am making a mistake, How do I use the condition please help me please.
Akash
Thanks
It is D) because $\sin^{-1}\left(\sin\frac{4\pi}{3}\right) = -\frac{\pi}{3}$, and $\cos^{-1}\left(\cos\frac{4\pi}{3}\right) = \frac{2\pi}{3}$, hence the result.